This paper deals with the stability and stabilization problems for a class of discrete-time nonlinear systems. The systems are composed of a linear constant part perturbated by an additive nonlinear function which satisfies a quadratic constraint. A new approach to design a static output feedback controller is proposed. A sufficient condition, formulated as an LMI optimization convex problem, is developed. In fact, the approach is based on a family of LMI parameterized by a scalar, offering an additional degree of freedom.
The problem of performance taking into account an

Modeling a real process is generally a complex and difficult task. Even if in numerous cases, a linear model can capture the main dynamical characteristics of a process. In some situations, it is necessary to take into account model uncertainties in order to design an efficient control law.

There exists an extensive literature dealing with this problem which is in fact the main problem in robust control design [

Among the numerous solutions allowing taking into account model uncertainties, a way which has been frequently investigated in literature consists in adding to the linear part of the model a nonlinear one which captures model uncertainties and frequently referred in literature as nonlinear systems with separated nonlinearity.

Nonlinear systems with separated nonlinearity are a class of nonlinear systems composed of a linear constant part to which another nonlinear function part is added. This function depends on both time and state and satisfies a quadratic constraint [

Many papers have investigated robust stability, analysis, and synthesis using essentially Lyapunov theory which has proved to be efficient in this context. Recent proposed approaches [

Even if the static output feedback stabilization (SOF) problem is considered as NP-hard [

The problem of performance is also treated in the context of

The paper is organized as follows. Section

For conciseness the following notations are used:

In this section, we consider nonlinear discrete-time system with the following state-space representation:

The nonlinear function

The parameter

The nonlinear function

In the sequel, we will use the following definition to present the concept of robust stability of the system (

System (

In this section, we develop a method for studying robust stability of system (

Let

See [

Given a symmetric matrix

Let

there exists a matrix

The proof is obtained remarking that (

We first introduce the following theorem which gives a robust stability conditions for system (

System (

See [

The stability condition given by Theorem

A new condition for robust stability is proposed in the following theorem.

System (

Inequality (

The two optimization problems (

In this section, we investigate the static output feedback stabilization problem for nonlinear discrete systems.

We consider the nonlinear discrete-time system described as follows:

The objective is to find a static output feedback control law such as

The closed loop system is given by the following state space representation:

Note that in this case, the constraint (

To establish a robust stabilization theory for system (

System (

According to the Theorem

Equation (

With this transformation we obtain the optimization problem given in Theorem

The optimization problem given by Theorem

In this section, we introduce a new approach for robust stabilization by SOF of nonlinear discrete time system (

The system (

According to Theorem

Then, inequality (

For this reason, we introduce some transformations to simplify the

With this transformation, we obtain the optimization problem given by Theorem

The following lemma gives a connection of the results of Theorem

If the SOF stabilization problem is solvable by Theorem

If we consider the optimization problem (

Stability is the minimum requirement and in practice a performance level has to be guaranteed. Performance objectives can be achieved via

We note

System (

Define the Lyapunov function:

By using the dissipative theory, we show that

Evaluating (

By Schur complement, (

Multiplying by

Now, we introduce the following theorem which can be seen as an alternate characterization of upper bounds of the

System (

Inequality (

In this section, we consider the static output feedback stabilization problem for the following nonlinear discrete system (

The system closed by SOF is written as

The objective of this section is to design static output feedback

System (

According to Theorem

In this paragraph, an

System (

According to the Theorem

We present in this section two numerical examples to illustrate the proposed theory for SOF synthesis.

Our approach Theorem

We consider the nonlinear discrete-time system of inverted pendulum (see Figure

The nonlinear system is unstable. The matrix

Numerical Evaluation for Example

Approach | ||
---|---|---|

Theorem | ||

Theorem |

Inverted pendulum scheme.

Output

Output

We consider now the nonlinear discrete-time system:

Numerical Evaluation for Example

Approach | ||

[ | 0.1954 | |

New approach, ( | 0.3084 | |

New approach, ( | 0.3329 |

We consider the nonlinear discrete-time system (

Numerical Evaluation for Example

Approach | ||

Approach, ( | 0.0032 | |

Approach, ( | 0.0288 |

Numerical Evaluation for Example

Approach | ||

Approach, ( | 0.3643 | |

Approach, ( | 0.4087 |

In this paper, the stabilization problem by static output feedback (SOF) for a particular class of nonlinear discrete time systems is investigated. A new sufficient condition is elaborated by using Lyapunov theory and formulated by LMI constraints. We obtain a convex optimization problem for maximizing the bound of the nonlinearity preserving the stability of the systems.

Finally, the proposed controller design method was extended to incorporate

Other classes of nonlinear discrete time nonlinear systems exist and present some interesting characteristic from a practical point of view. For example, the ones where the linear part is affected by polytopic uncertainties. It would be interesting to extend the results of this paper to those classes. This will be exploited in a near future.